The diffusive limit of Boltzmann equation in torus

The Boltzmann equation of kinetic theory gives a statistical description of a gas of interacting particles. It is well known that the Boltzmann equation is related to the Euler and Navier–Stokes equations in the field of gas dynamics. In this paper we are concerned with the incompressible Navier–Stokes–Fourier limit of the Boltzmann equation. We prove the incompressible Navier–Stokes–Fourier limit globally in time and the time decay rate of the solution to the rescaled Boltzmann equation in a torus. For ɛ small, by using the truncated expansion and L x , v 2 – L x , v ∞ method, we prove such a limit for the general potentials γ ∈ ( − 3 , 1 ] .